358 XV NORMED ALGEBRAS AND SPECTRAL THEORY Remark (15.7.4.1) If E is a Hilbert space of finite dimension «, we can define a scalar product (u\v) on «S?(E) = Endc(E) as in (15.7.4). As in (15.4.8) we see that, if we put \\u\\l = (u\ u), then \\u\\ 2 is a norm on JSf(E) such that \\uv\\2 ^ \\u\\2 • \\v\\2l it is necessarily equivalent to the usual norm \\u\\ (5.9.1), but does not satisfy (15.1.2) unless n = 1, because ||1E||2 = n112. (15.7.5) Let A be an arbitrary algebra with involution, and let g be a bitrace on A. Since g(s*, s-*) = g(s, s), the left ideal n defined in (1 5.6.8) is self- adjoint and is therefore a two-sided ideal. The vector space A/n is therefore naturally an algebra. Since n is self-adjoint, the mapping si->s* induces an involution on A/n such that n(s*) = (n(s))* ; also the scalar product (jc | y) on A/n induced by g is a bitrace on A/n ; finally, if # satisfies the condition (U) (resp. (N)), then the same is true of the scalar product on A/n. A Hilbert algebra is defined to be an algebra A with involution, endowed with the structure of&prehilbert space defined by a bitrace (x \ y) satisfying the conditions (U) and (N). In other words, this scalar product satisfies the follow- ing conditions : (15.7.5.1) ( (15.7.5.2) (xy\z) = (y\x*z) (15.7.5.3) For each x e A there exists a real number M^ ^ 0 such that (15.7.5.4) The elements of A of the form xy, -with x e A and y e A, form a total set in A. Note that it follows from (15.7.5.1) and (15.7.5.2) that (15.7.5.5) For (yx \ z) = (x*y* \ z*) = (y* \ xz*) = (y \ zx*). Furthermore, we have 0 5-7.5.6) (yx | yx) ^ Mx,(y \ y) because (yx \ yx) = (x*y* \ x*y*) ^ Mx,(y* \ y*) =4.8) and that if w(E) = F is finite-